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  • $\begingroup$ In the third paragraph you first write $\mu(B)=\epsilon$, and immediately after $\mu(B)>\epsilon$. I think you want the first equality to be an inequality as well. $\endgroup$ Commented Jan 8, 2019 at 15:50
  • $\begingroup$ @Wojowu Yeah, I caught that at the same time. $\endgroup$ Commented Jan 8, 2019 at 15:53
  • $\begingroup$ Beautiful @NoahSchweber - and lightning fast! $\endgroup$ Commented Jan 8, 2019 at 15:54
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    $\begingroup$ Not a fascinating remark, but the proof also holds with a limsup in the definition of $\mu$. $\endgroup$ Commented Jan 8, 2019 at 16:34
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    $\begingroup$ @PierrePC Indeed, and that's actually how I (mis)read the question. :P $\endgroup$ Commented Jan 8, 2019 at 17:15